2,036 research outputs found
Reflections on Tiles (in Self-Assembly)
We define the Reflexive Tile Assembly Model (RTAM), which is obtained from
the abstract Tile Assembly Model (aTAM) by allowing tiles to reflect across
their horizontal and/or vertical axes. We show that the class of directed
temperature-1 RTAM systems is not computationally universal, which is
conjectured but unproven for the aTAM, and like the aTAM, the RTAM is
computationally universal at temperature 2. We then show that at temperature 1,
when starting from a single tile seed, the RTAM is capable of assembling n x n
squares for n odd using only n tile types, but incapable of assembling n x n
squares for n even. Moreover, we show that n is a lower bound on the number of
tile types needed to assemble n x n squares for n odd in the temperature-1
RTAM. The conjectured lower bound for temperature-1 aTAM systems is 2n-1.
Finally, we give preliminary results toward the classification of which finite
connected shapes in Z^2 can be assembled (strictly or weakly) by a singly
seeded (i.e. seed of size 1) RTAM system, including a complete classification
of which finite connected shapes be strictly assembled by a "mismatch-free"
singly seeded RTAM system.Comment: New results which classify the types of shapes which can
self-assemble in the RTAM have been adde
Force induced triple point for interacting polymers
We show the existence of a force induced triple point in an interacting
polymer problem that allows two zero-force thermal phase transitions. The phase
diagrams for three different models of mutually attracting but self avoiding
polymers are presented. One of these models has an intermediate phase and it
shows a triple point but not the others. A general phase diagram with
multicritical points in an extended parameter space is also discussed.Comment: 4 pages, 8 figures, revtex
Full Disclosure, Market Discipline, and Risk Taking: Rethinking Confidentiality in Bank Regulation
Constructive Dimension and Turing Degrees
This paper examines the constructive Hausdorff and packing dimensions of
Turing degrees. The main result is that every infinite sequence S with
constructive Hausdorff dimension dim_H(S) and constructive packing dimension
dim_P(S) is Turing equivalent to a sequence R with dim_H(R) <= (dim_H(S) /
dim_P(S)) - epsilon, for arbitrary epsilon > 0. Furthermore, if dim_P(S) > 0,
then dim_P(R) >= 1 - epsilon. The reduction thus serves as a *randomness
extractor* that increases the algorithmic randomness of S, as measured by
constructive dimension.
A number of applications of this result shed new light on the constructive
dimensions of Turing degrees. A lower bound of dim_H(S) / dim_P(S) is shown to
hold for the Turing degree of any sequence S. A new proof is given of a
previously-known zero-one law for the constructive packing dimension of Turing
degrees. It is also shown that, for any regular sequence S (that is, dim_H(S) =
dim_P(S)) such that dim_H(S) > 0, the Turing degree of S has constructive
Hausdorff and packing dimension equal to 1.
Finally, it is shown that no single Turing reduction can be a universal
constructive Hausdorff dimension extractor, and that bounded Turing reductions
cannot extract constructive Hausdorff dimension. We also exhibit sequences on
which weak truth-table and bounded Turing reductions differ in their ability to
extract dimension.Comment: The version of this paper appearing in Theory of Computing Systems,
45(4):740-755, 2009, had an error in the proof of Theorem 2.4, due to
insufficient care with the choice of delta. This version modifies that proof
to fix the error
Dynamics and Transport in Random Antiferromagnetic Spin Chains
We present the first results on the low-frequency dynamical and transport
properties of random antiferromagnetic spin chains at low temperature (). We
obtain the momentum and frequency dependent dynamic structure factor in the
Random Singlet (RS) phases of both spin-1/2 and spin-1 chains, as well as in
the Random Dimer phase of spin-1/2 chains. We also show that the RS phases are
unusual `spin-metals' with divergent low-frequency conductivity at T=0, and
follow the spin conductivity through `metal-insulator' transitions tuned by the
strength of dimerization or Ising anisotropy in the spin-1/2 case, and by the
strength of disorder in the spin-1 case.Comment: 4 pages (two-column format). Presentation substantially revised to
accomodate new result
The Power of Duples (in Self-Assembly): It's Not So Hip To Be Square
In this paper we define the Dupled abstract Tile Assembly Model (DaTAM),
which is a slight extension to the abstract Tile Assembly Model (aTAM) that
allows for not only the standard square tiles, but also "duple" tiles which are
rectangles pre-formed by the joining of two square tiles. We show that the
addition of duples allows for powerful behaviors of self-assembling systems at
temperature 1, meaning systems which exclude the requirement of cooperative
binding by tiles (i.e., the requirement that a tile must be able to bind to at
least 2 tiles in an existing assembly if it is to attach). Cooperative binding
is conjectured to be required in the standard aTAM for Turing universal
computation and the efficient self-assembly of shapes, but we show that in the
DaTAM these behaviors can in fact be exhibited at temperature 1. We then show
that the DaTAM doesn't provide asymptotic improvements over the aTAM in its
ability to efficiently build thin rectangles. Finally, we present a series of
results which prove that the temperature-2 aTAM and temperature-1 DaTAM have
mutually exclusive powers. That is, each is able to self-assemble shapes that
the other can't, and each has systems which cannot be simulated by the other.
Beyond being of purely theoretical interest, these results have practical
motivation as duples have already proven to be useful in laboratory
implementations of DNA-based tiles
Bounding the dimensions of rational cohomology groups
Let be an algebraically closed field of characteristic , and let
be a simple simply-connected algebraic group over that is defined and
split over the prime field . In this paper we investigate
situations where the dimension of a rational cohomology group for can be
bounded by a constant times the dimension of the coefficient module. We then
demonstrate how our results can be applied to obtain effective bounds on the
first cohomology of the symmetric group. We also show how, for finite Chevalley
groups, our methods permit significant improvements over previous estimates for
the dimensions of second cohomology groups.Comment: 13 page
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Interconnected Self-Propagating Photopolymer Waveguides: An Alternative to Stereolithography for Rapid Formation of Lattice-Based Open-Cellular Materials
Recently, a new technique has been developed to create unique open-cellular materials with
micro-scale truss, or lattice features ranging from tens to hundreds of microns. These materials
are formed from a three-dimensional, interconnected array of self-propagating photopolymer
waveguides. By utilizing this self-propagating effect, three-dimensional open-cellular polymer
materials can be formed in seconds. In addition, intrinsic to the process is the ability to control
specific micro-lattice parameters which ultimately affect the bulk material properties. Unlike
stereolithography, this new fabrication technique is rapid (~ minutes to form an entire part) and
relies on a single two-dimensional exposure surface to form three-dimensional structures
(thickness > 25 mm possible). This combination of speed and planar scalability opens the
possibility for large-scale mass manufacturing. The utility of these new materials range from
lightweight energy absorbing structures to thermal management materials to bio-scaffolds.Mechanical Engineerin
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